Statistics of Mars’ Topography from the Mars Orbiter LaserAltimeter: Slopes, Correlations, and Physical Models

Oded Aharonson, Maria T. Zuber�, and Daniel H. Rothman

Department of Earth, Atmospheric and Planetary Sciences,Massachusetts Institute of Technology, Cambridge, Massachusetts

Abstract. Data obtained recently by the Mars Orbiter Laser Altimeter was usedto study the statistical properties of the topography and slopes on Mars. We findthat the hemispheric dichotomy, manifested as an elevation difference can bedescribed by long baseline tilts, but in places is expressed as steeper slopes. Thebimodal hypsometry of elevations on Mars becomes unimodal when referencedto the center of figure, contrary to the Earth for which the bimodality is retained.However, ruling out a model in which the elevation difference is expressed ina narrow equatorial topographic step cannot be done by the hypsometry alone.Mars’ slopes distribution is longer tailed than that of Earth and Venus, indicatinga lower efficiency of planation processes relative to relief building tectonics andvolcanics. We define and compute global maps of statistical estimators includingthe inter-quartile scale, RMS and median slope, and characteristic decorrelationlength of the surface. A correspondence between these parameters and geologicunits on Mars is inferred. Surface smoothness is distinctive in the vast northernhemisphere plains, where slopes are typically � 0.5 � . Amazonis Planitia exhibitsa variation in topography of � 1 m over 35 km baseline. The region of hematitemineralization in Sinus Meridiani is also smooth with median slopes lower than����� � , but does not form a closed basin. The shallower long wavelength portionof the lowlands’ topographic power spectrum relative to the highlands’ can beaccounted for by a simple model of sedimentation such as might be expected atan ocean’s floor. The addition of another process such as cratering is necessaryto explain the spectral slope in short wavelengths. Among their application, theseMOLA-derived roughness measurements can help characterize sites for landingmissions.

Introduction

As descriptors of planetary surfaces, slopes and slope dis-tributions are pertinent to the mechanisms of formation ofphysiographic features, and are indicative of the style andduration of subsequent modificational processes [e.g. Schei-ddeger, 1991]. As a step towards quantifying the nature ofsurface processes of Mars, we analyze elevation and slopestatistics derived from profiles collected by the Mars OrbiterLaser Altimeter (MOLA) [Zuber et al., 1992], an instrumenton the Mars Global Surveyor (MGS) spacecraft. Since the

�Also at: Laboratory for Terrestrial Physics, NASA/Goddard Space

Flight Center, Greenbelt, Maryland.

origin of some large and small scale surface features remainsin debate, the task of interpreting statistical models for geo-physical information is ambiguous. Nonetheless, such sta-tistical estimation can be useful, especially in a comparativesense and when additional observation types are includedin the interpretation. For example, the topography of theEarth’s seafloor has been characterized in detail by severalstudies [Fox and Hayes, 1985; Goff and Jordan, 1988; Neu-mann and Forsyth, 1995; Smith and Jordan, 1988] in termsof models describing its statistical properties and powerspectrum, and several processes have been modeled withthese tools including seafloor spreading, sedimentation, andseamount distribution. The statistics of continental topogra-

1

2 AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY

phy has been studied in landscapes such as eroding environ-ments and river networks (e.g. in Rodriguez-Iturbe and Ri-naldo [1997]; Scheiddeger [1991]). The stereographically-derived topographic field of Mars that predated MGS [Wu,1991; Esposito et al., 1992] permitted only coarse study dueto its low precision and resolution. With the newly obtainedinformation, an accurate statistical characterization is finallypossible. Results based on initial data collected in the north-ern hemisphere were reported on in Aharonson et al. [1998],and more globally in Aharonson et al. [1999], Kreslavskyand Head [1999] and Kreslavsky and Head [2000].

Data Collection and Accuracy

The data set on which the following analysis is based wascollected during the aerobreaking hiatus, science phasing,and ongoing mapping orbits of Mars Global Surveyor [Al-bee et al., 1998, 2000]. The data provide coverage of thesurface of Mars at 300-400 m spacing along track as dic-tated from the spacecraft orbital velocity and the laser pulserepetition frequency of 10 Hz. The MOLA instrument has arange resolution of 37.5 cm, a range precision of 1-10 m forsurface slopes of up to 30 � , and an absolute accuracy of � 1m with respect to Mars’ center of mass [Zuber et al., 1992;Neumann et al., 2000]. These data currently constitute thehighest quality measurements of Mars topography and per-mit quantification of slopes from local to hemispheric scale[Smith et al., 1999, 2001]. The accuracy of point-to-pointslopes along track is

������� � . Resolving steep slopes is lim-ited by the instrument’s detection of reflected power, but thisis rarely encountered, and only at slopes greater than ��� . Inthe following sections various statistical estimators will bedefined and applied to the data set. A global view of eleva-tions and slopes is presented first, followed by regional slopeand power spectral characteristics.

Global Hypsometry

The most striking global feature of the surface of Marsis the crustal dichotomy. The northern hemisphere is topo-graphically lower, morphologically smoother, and geolog-ically younger than the south [Carr, 1981; Mutch et al.,1976]. A frequency diagram of elevations, or hypsogram(Figure 1a) shows this dichotomy clearly, as was previouslyobserved [e.g. Smith and Zuber, 1996; Smith et al., 1999].The strong bimodal distribution shows the lowlands peak is� 5.5 km lower than the highlands, and is narrower, indicat-ing the flatness of the north. The distribution’s lowest por-tion, the peak between �� km and ��� km, is the floor ofthe Hellas impact basin in the south. Sharpton and Head[1985] and Sharpton and Head [1986] compared the topog-raphy and slopes of Earth and Venus. Using the same bin

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Figure 1. Hypsometry of surface topography from griddedelevation models. Bin width is 100 m. To compute an areaweighted histogram an equal-area projection of the data wasused. The black curve is the elevation histogram with respectto the geoid, and the gray filled area is the distribution withthe degree 1 terms removed. The distributions shown are for(a) Mars, (b) a synthetic “noisy cliff model”, and (c) Earth.

widths, the hypsometry of Mars has a larger overall range ofelevations than Earth (and Venus), and is bimodal, appearingsimilar to Earth shown in Figure 1(c) [NGDC, 1988]. How-ever, as shown in Smith et al. [1999] and by the solid blackcurves, by removing the spherical harmonic degree 1 terms,effectively referencing the topography to the geometric cen-ter of figure (COF) instead of the center of mass (COM), thedistribution of topography of Mars becomes unimodal (as isVenus), whereas Earth maintains its bimodality.

It is instructive to consider a theoretical planet whose to-pography consists of an equatorial scarp of height 6 km with1 km of normally distributed noise superimposed. Figure1(b) shows the corresponding distributions of this “noisycliff” model. Referencing to the COF also collapses the twosharp peaks of this model into one, but depending on the

AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY 3

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Figure 2. A histogram of regional slopes for Mars (solidcurve), Earth (dashed) and Venus (dotted). Baselines forslope calculations were 100 km, bin width is 0.01 � , similarto Sharpton and Head [1985]. Note the relative abundanceof steeper slopes in the range 0.18-0.42 � on Mars.

amount of noise, shoulders can remain in the histogram. Wetherefore conclude that the elevation histogram of Mars isconsistent with the dichotomy having a global effect, but anequatorial scarp is not ruled out by the hypsometry alone.Additional analysis is required to isolate contributions tothe hemispheric elevation difference, for example by crustalthickness variations, as in Zuber et al. [2000b].

It is also possible to compare regional slopes on Marsto Sharpton and Head [1985] results for Earth and Venus.Figure 2 shows an equal area histogram of 100-km slopesbinned in 0.01 � bins. Similar to Venus, the distribution forMars peaks near 0.05 � , but is longer tailed. In fact, on Venusregional slopes rarely exceed 0.3 � . The steeper slopes onMars are presumably related to the lower efficiency of pla-nation processes (which on Earth are often driven by the hy-drosphere and atmosphere). However, the expected value ofthe absolute slope varies greatly by region, and will be dis-cussed in later sections.

Long-Baseline Tilts

Several studies have demonstrated that Mars’ topographyis characterized by a hemispherical-scale tilt. For exam-

ple, Smith et al. [1999] and Zuber et al. [2000a] observedthat in a longitude band near the Mars’ prime meridian, thepole-to-pole topography can be described to first order by apole-to-pole slope of � 0.036 � downhill to the north. In ad-dition Phillips et al. [2000] consider planetary-scale slopesdue to Tharsis. To quantify long-baseline slopes on Mars,we gloablly compute slopes on 1000 km baselines by fittingplanes of that dimension to locally-projected gridded fields.The MOLA-derived topographic field has an absolute accu-racy of � 1 m with respect to Mars’ center of mass. Even al-lowing for errors in the height of the geoid of � 5 m [Lemoineet al., 2000], 1000 km surface slopes have precision of betterthan � 2 ��� � ����� . Plate 1 shows the magnitude (color-coded)and direction (arrows) of these regional slopes. It is clear thatthe majority of the northern hemisphere slopes downwardsto the north. The lowlands have a typical tilt of 0.02-0.1 �with values increasing to 0.4 � near the dichotomy bound-ary in longitudes 50-210 � E and the large volcanos. Thebasin floor of Hellas (44.3 � S 66.2 � E) has an overall east-erly tilt, although locally it has some northerly slopes. How-ever, the basin’s topography has been significantly alteredsince its formation [Wichman and Schultz, 1989; Tanaka andLeonard, 1995], so it is difficult to ascertain its primordialshape. The floor of the Utopia basin (47.6 � N 82.7 � E) showsno discernible non-radial tilts by this method, and attemptsto reproduce the northerly-tilt result of McGill [2001] whereunsuccessful. Naturally, the north polar cap is tilted ap-proximately south, while the south cap is tilted north, giv-ing a quasi-parabolic shape which is consistent with ablation[Zwally et al., 2000] and glacial flow [Zuber et al., 1998b, a]models.

Arabia Terra ( � 30 � N 30 � E) is broadly tilted by 0.05-1.5 �to the north-west, a direction to which many of the channelsin the region closely adhere. This correspondence is alsotrue in the area of Chryse (27 � N 324 � E), although the slopesare about 0.05 � steeper. The correlation between channeldirection and gradient azimuth as a function of scale of thelocal slope is an exciting research direction presently beingpursued in greater detail. It is notable here that channels areoften oriented in a direction consistent with long-baselineslopes.

Regional trends were also used in an analysis by Mouginis-Mark et al. [1982], where slope azimuths were comparedto flow directions of Tharsis lava flows, that were taken aspaleo-slope indicators. They found almost no deformationassociated with loading since the flow emplacement. Themore precise slopes computed here are similar to the onesthey used, and therefore confirm their conclusion that duringthe current epoch of preserved volcanic activity the litho-spheric thickness must be large (perhaps � 150 km). Thisscenario is consistent with the thick elastic lithosphere de-

4 AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY

termined from gravity and topography inversions of Zuberet al. [2000b].

The orientation of the long-baseline slopes in the south-ern hemisphere is complicated by large impact and vol-canic constructs such as Hellas, Argyre and Tharsis (Fig-ure 3 (b,d), Plate 1). At regions distant from these features,for example, near longitudes 0 � and 170-180 � E (Figure 3(a,c), Plate 1) there slopes appear to be biased towards thenorth. Still other longitude bands (e.g. � 130 � E, 200-210 �E) have random distribution of slope azimuths. The com-puted slopes indicate that a substantial fraction of the planetis tilted roughly northwards, altough important departuresfrom that exist, for instance near Tharsis and Hellas. Thissupports the long-held view that the majority of the sur-face of Mars drains north [Banerdt, 2000], and the lowlandsare a natural sink for not only volatiles, but also sedimen-tary, aeolian and volcanic materials [Smith et al., 2001]. Al-though a description of long-baseline slopes is provided bythe method presented, the question of possible global tiltsremains open and attractive to further investigation using thehighly precise MOLA data set.

RMS and Median Slope

Short wavelength slopes are relevant to regional scaleprocesses on Mars. In order to investigate short baselineslopes, MOLA tracks were numerically differentiated by a3-point Lagrange formula:

�������� ��� � � ��� � ��

�������� � ���� � � �� � � ���������� �� � � �� � ���

� ���� �� � (1)

for three consecutive points at ��� �� � ������� with !�"� � � � � �and

��$# � �� � �%#. For equally spaced points, this is equiv-

alent to a centered difference scheme. Formally, the resultshould be viewed as a lower limit on the absolute slope sincethe measurement is restricted to the ground-track directionwhich may not follow the gradient direction. If the topogra-phy field is isotropic, the average properties that follow caneasily be corrected for this, by dividing by the average valueof &('*),+*��- � & , that is ,.�/ .

It is customary to express surface roughness in terms ofthe root mean-squares (RMS) of slopes -10 , because radarreflection scatter is largely affected by this parameter. How-ever, a typical slope distribution need not, and in fact rarelydoes, resemble a Gaussian, which is the distribution forwhich RMS is the appropriate measure. Point-to-point MOLAslopes can be anomalously large (e.g. due to small cratersor faults) and contribute to a long-tailed distribution. The

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Figure 3. Pole-to-pole topographic profiles stacked in se-lected 10 � longitude intervals.

L2 RMS slope estimate is sometimes dominated by thesehigh slopes, and is therefore not always representative of thetypical nature of the surface [Neumann and Forsyth, 1995;Kreslavsky and Head, 1999].

Nonetheless, empirical study shows a surprising agree-ment between some RMS slope estimates derived from radarobservations and from MOLA [Aharonson et al., 1999].Hence RMS slopes are reported here, but with a cautionarynote especially when these values are high.

RMS slopes were computed in a 35-km running win-dow. The approximately hemispheric crustal dichotomy ismanifested, with the southern hemisphere having typicalRMS slopes in the range 3-10 � and the northern hemispherein the range 0.2-0.8 � [Head et al., 1999]. Exceptions tolow slopes in the north include the Olympus Mons (18.4 � N226.9 � E) basal escarpment, canyon walls, sparse craters, andthe edges of the ice cap deposits, while in the south lowslopes are evident in the relatively flat portions of the icecap deposits.

Radar observations of Mars [Harmon et al., 1999; Chris-

AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY 5

tensen, 1986; Christensen and Moore, 1992] yield values for-�0 in the range

� � � � � � � , very similar to the overall rangeobtained here. The Tharsis Montes and Olympus Mons areasexhibit relatively high -10 of � 10 � in radar, and the MOLAdata agrees well. The Amazonian region west of Tharsis(� � � N ��� � E) has large -�0 in radar, but appears remark-

ably smooth in MOLA data. This indicates that althoughthis region appears very smooth at MOLA scale, it may berougher at a smaller scale. Chryse Planitia ( � N � � E),whose radar appearance played a role in site selection forViking Lander 1, has -�0 of �� � ��� in radar, and similarlyin MOLA in the southern portion with values decreasing to� 1 � in inter-crater plain to the north. Syrtis Major at

� � � N � � E has been observed to have some of the lowest effective-�0 in radar of � 1 � down to

� � � , and shows a slightly largerRMS slope by MOLA. This is an example of a surface that isprobably smoother on a scale smaller than MOLA samplingthan the point-to-point slopes would imply. Finally, the northpolar region has a uniform radar response of - 0 � 2 � , in fairagreement with MOLA.

In characterizing typical surface gradients, median abso-lute slopes were found to be most robust. This statistic isleast sensitive to the choice of windows, and adjacent orcrossing tracks show nearly identical values. While RMSslopes suffer when distributions are long tailed due to theaveraging of squares of slopes, median absolute slopes areessentially unaffected by the height of the distribution’s tail[Kreslavsky and Head, 1999]. Plate 2 shows maps of medianslopes in the same format as before.

A comparison with RMS slopes indicates that the medianslope values are generally smaller, consistent with the re-marks stated previously, that a small amount of steep slopescan dominate the RMS measure. It is observed that me-dian slopes contrast surfaces distinctly, so that for examplethe dune-covered Olympia Planitia ( � � N � � � � E) appearsslightly rougher than the north polar cap margins. Impactbasins floor and rim characteristics emerge prominently. Thelarge craters have a common morphology in which the floorsare generally smoother than the rims and associated ejectablankets. An interesting boundary in roughness occurs at thenorthern margin of the Tharsis province, approximately con-centric with Alba Patera (40.5 � N 250.1 � E) but � 1200 kmnorth of its center. Median slope values drop from � 0.25 � onthe volcano flank side to

�0.15 � on the lowlands side. The

roughness increases again to the north, forming a crescent-shape region smoother than its surroundings, perhaps relatedto a volcanic episode.

Another area of particular geological interest in SinusMeridiani. Recently, probable hematite mineralization hasbeen identified there [Christensen et al., 2000] by the Ther-mal Emission Spectrometer (TES), an instrument also on

board MGS. In plate 3 topography and median slope mapsfor this region are shown, as well as the mapped distributionof hematite from [Christensen et al., in press]. The presenceof hematite closely correlates with a smoother region of me-dian slopes

�0.4 � . However the region appears to have a

south-westerly slope, and is not a closed depression. Chris-tensen et al. [2000] examined two classes of hematite for-mation mechanisms: one that requires significant amountsof near surface water, and one that does not. The unusualsmoothness, if a result of sub-aqueous sedimentation (whicharguably is speculative, but is often associated with iron inthe presence of standing bodies of water on Earth), clearlyfavors the first explanation over the latter.

In summary, the measured median slopes are boundedat the lowest values by Amazonis Planitia and some polarcap deposits (further discussion of this will follow). At thehighest variation of slope, in order of increasing median, arethe Hellas rim, Argyre rim, dichotomy boundary (especiallyeast of � � � E), Olympus Mons aureole, and canyon walls ofValles Marineris, that are in excess of � � .

Inter-Quartile Scale

It is often useful to consider roughness in terms of typicalrelative elevation deviations, rather than slope. One way toachieve this to employ the inter-quartile scale (IQS). In thischaracterization we measure the width of a histogram of the ��� most significant elevations, scaled to unity for a nor-mal distribution. Before normalization, this estimator ��� isdefined Neumann and Forsyth [1995] as

� � ��

� � � ������� � �(2)

where � � is the elevation of the � �� quartile point and�

isthe number of points. To normalize, � � is divided by 0.673,the IQS of a normal distribution. The parameter ��� is a ro-bust estimator in the sense that it is not sensitive to outliers inas much as half of the population. This calculation is appliedto all range returns that fall within a window sliding alongeach profile. Although choice of window size can affect thenumerical values measured, the qualitative results below re-main true. In its rank 1 statistical nature, IQS is comparableto median slope except that it operates directly on elevationsrather than elevation difference (slope). To avoid a regionalslope bias, a mean trending surface is removed prior to thecalculation. The results, computed in 35-km windows, areplotted in Plate 4. One advantage of IQS measurements isthat they can be directly compared to vertical roughness asdetermined from MOLA pulse width measurements [Garvinand Frawley, 2000]. The pulse width is sensitive to rough-ness on smaller scales, up to the laser spot size of � 100 m

6 AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY

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Plate 1. Long baseline (1000 km) tilts. Azimuths are indicated by arrows, and magnitude by color. The fields are in Mercator(a) and Stereographic (b,c) projections over the poles. A north-easterly illuminated shaded relief model of the topography issuperimposed in monochrome. Note the northerly tilt of much of the northern hemisphere.

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AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY 7

Table 1. Data sets used in the planetary surface comparion of Figure 4, and their approximate horizontal resolution andvertical accuracy.

Planet Instrument / Source Region Accuracy Resolution Data Reference

Mars MOLA Amazonis Planitia 0.4 m 0.3 km [Smith et al., 2001]Moon Clementine Oceanus Procelarrum 40 m 0.2 km � [Smith et al., 1997]Venus Magellan Niobe Planitia 4 m 10 km [Ford and Pettengill, 1992]Earth SLA Sahara Desert 1.5 m 0.7 km [Garvin et al., 1997]Earth Seabeam 2200 South Atlantic 2 m 0.1 km [Neumann et al., 1996]Earth GTOPO DEM Great Plains, US 20 m

�0.1 km [Gesch and Larson, 1996]

Earth GTOPO DEM Tibetan Plateau and basins 20 m�

0.1 km [Gesch and Larson, 1996]

� resolution is � � � � km for 1 Hz data and � � � km for 8 Hz data�highly variable

Figure 4. Comparison of planetary surface topography(From Aharonson et al. [1998]). The data sets used are listedin Table 1.

on the ground. Comparison with the vertical roughness re-ported in Smith et al. [2001] shows good relative agreement,with the absolute values systematically greater for the largerscale IQS.

Some observational examples of the IQS measurementsover various geological provinces are presented below. Manyqualitative observations also appear in the median slopescharacterizations.

The northern hemisphere is flat, with typical inter-quartilescale of a few tens of meters ranging over thousands of kilo-meters [Aharonson et al., 1998]. The Olympus Mons aure-ole deposits are among the roughest surfaces observed, witha typical IQS of hundreds of meters. Compared with thesurrounding terrain and the rest of the southern hemispherewhere typical IQS is � 30 m, the south-west portion of theHellas impact basin rim appears smoother, with an IQS de-creasing to � 10-30 m and extending essentially to the pole.This smoothness has been suggested to be a result of inter-action with surface ice [Kargel and Strom, 1992], perhapsrelated to the nearby south polar cap.

The most unusual region is Amazonis Planitia, an areato the northwest of Olympus Mons of Amazonian age andelevation of approximately -4.1 km. This surface displaysan IQS variation in topography of only a few meters (stillgreater than the instrument’s range resolution), extendingover hundreds of kilometers, and correlating well with thepreviously-mapped geology. The smoothest part of the sur-face corresponds to member 3 of the Arcadia Formation,which was previously interpreted to consist largely of lavaflows and small volcanos [Scott and Tanaka, 1986; McEwenet al., 1999]. Member 3 also forms smooth plains west of theOlympus Mons aureoles and displays occasional flow frontsin Viking images. This area has an anomalously low ther-

8 AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY

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Plate 3. The Sinus Meridiani region. Shown are (a) topog-raphy, (b) median slopes, and (c) Hematite distribution fromChristensen et al. [in press] plotted over an image mosaic ofthe region, with the

� � � median absolute slope contour alsoindicated.

mal inertia (between 2 and 3 � � � � cal cm�

s� ����

K� �

[Christensen and Moore, 1992]) with very low variabilityover tens of km scale and anomalously low radar backscat-ter cross-section at a variety of wavelengths [Jakosky andMuhlemann, 1981]. It has more recently been interpreted tobe accumulations of fine-grained dust [Christensen, 1986].The regional flatness and low surface roughness detectedby MOLA is consistent with this interpretation but does notuniquely explain the genesis of this region.

An initial step towards identifying the mechanism of for-mation of Amazonis Planitia is to compare its topographicproperties to other smooth regions with potentially analo-gous origins. Shown in Figure 4 are profiles of elevation col-lected by various altimeters over smooth surfaces from a va-riety of solar system bodies. The data sets used are listed inTable 1. At the top is MOLA Pass 31 over Amazonis Plani-tia, where the anomalously smooth region is observed to ex-tend over 600 km, approximately centered in the plot. Be-low is a Clementine profile of the Moon’s Oceanus Procel-larum, Magellan radar altimetry over one of the smoothestareas of Venus, Niobe Planitia, Shuttle Laser Altimeter datacollected over the Sahara desert, and shiptracks of Seabeam2200 bathymetry over the south Atlantic abyssal plains. Thelast two profiles were extracted from the GTOPO data set,first over the Great Plains in the U.S., and second over theIndo-Gangedic Plains across over the Tibetan Plateau, downacross the Tarim Basin and continuing northwards. OceanusProcellarum consists of lava flows that have been broadlytilted by subsidence and locally steepened by tectonic defor-mation (wrinkle ridges); their small-scale roughness is dom-inated by impact regolith formation processes. Niobe Plani-tia on Venus consists of vast lava plains similarly tilted andsteepened but not influenced by regolith formation [Tyleret al., 1992]. Comparison of these surfaces reveals that ofthese lowest, smoothest regions observed in the solar system,Amazonis Planitia closely resembles in its smoothness onlythe heavily sedimented surfaces on the Earth, i.e. oceanicabyssal plains and basins filled by fluvial deposition pro-cesses. It is noteworthy that volcanically-resurfaced terrainis markedly rougher on the Moon, on Venus, and on Mars,than the peculiar Amazonis deposits. Saharan sand sheetsare rougher in terms of IQS by a factor of about three. Otherregions in the Martian northern hemisphere that exhibit evi-dence of dust deposition are rougher than Amazonis as well.Areas similar in size and IQS are found on Mars only at thenorth and south ice covered caps.

Horizontal Decorrelation Length

Thus far the vertical component of the topography hasbeen characterized by various statistical estimators. One

AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY 9

way to define a characteristic horizontal length-scale is byestimating the distance over which the topography first decor-relates with itself. This distance is obtained by comput-ing the auto-covariance function � of a sample population� � � � � � ��� � � ��� � as a function of the lag � , and measuringits width at half-maximum. The auto-covariance function isestimated by:

� ����� � ���

���� � � � �

��� � � � �

���� �� �� (3)

where � is the sampling interval,�

is the number of sam-ples in the window, and � is the mean of

�. The length � ,

given by � ��� � � � � � � � � (4)

measures the width of the central peak in the auto-covariancefunction. Hence in the ensuing discussion, � as defined in(4), will be referred to as a decorrelation length. Other defi-nitions are possible. The conventional definition for the cor-relation length can be used to estimate ��� with

� � ����� � �� � � � � � � �

��

� �� � � � � ��� �

(5)

Here we measure � , rather than � � , because � does not dependon the behavior of � at large lags. The value of � still dependson window size, but its relative variations over the Martiansurface do not. In the following analysis 35-km windowswere employed, as before.

Plate 5 shows a map view of the decorrelation length.Several interesting trends emerge. First, a global equator topole increase in decorrelation length is visible, from � 1000m at the equator to � 1800 m near both poles. Kreslavskyand Head [2000] observed a similar latitudinal trend in theirroughness estimator and suggest it is related to seasonal de-position/sublimation cycles or alternatively to terrain soften-ing effects (also in Garvin et al. [2000]). In the northernhemisphere there is an additional drop in � to � 500 m at sev-eral locations (Utopia Planitia, Isidis, northern Chryse Plani-tia), and a unique drop to � 250 m at Amazonis Planitia. Byinterpolating the auto-covariance function, even shorter val-ues for � are observed distinctly at the polar dune fields es-pecially in Olympia Planitia ( � � N � � � � E). Values of � thatare smaller than the sampling distance ( � � � � m) should betreated with extra caution, because they rely on an extrapo-lation beyond the shortest wavelength observed.

Power Spectral Density

One of the most intriguing hypotheses that has arisenbased on data from MGS is the possibility of an ancient

liquid water ocean residing in the northern lowlands [Headet al., 1998, 1999]. The relative smoothness of the lowlands,it has been suggested, resulted from sedimentation processesthat are expected on an ocean floor. In this section one testof this hypothesis is considered.

Two areally large regions on Mars have been selected torepresent the heavily cratered terrain in the south (Region A)and the smooth terrain in the north (Region B). Region A isbetween latitude 60 � S-30 � S and longitudes 150 � E-210 � E.This is the most ancient, heavily cratered crust on Mars,where strong magnetic anomalies have been observed. Re-gion B surrounds the planet in longitude and is bounded bylatitudes 60 � N-75 � N.

The large number of MOLA profiles allows a highlyaccurate estimation of the power spectral density function(PSD). In Figure 5 the means of many hundreds of 1-Dspectra along MOLA profiles in these regions are plotted.If the spectrum is roughly divided in two, a short wave-length portion (0.7

��� �7 km), and a long wavelength por-

tion (20��� �

200 km), then it is seen that although the mag-nitude of the power in short wavelengths is smaller in regionB than in region A, the slope is almost unchanged. Howeverthe slope of the long wavelength portion is substantially re-duced in region B relative to A.

There is a large body of work on descriptions of scal-ing relationships in terms of geomorphological models (re-viewed in Dodds and Rothman [2000]). One simple modelfor the complex process of sedimentation consists of a diffu-sion process forced with random noise [Edwards and Wilkin-son, 1982]. Figure 6 illustrates a surface evolving underthese conditions. The cartoon shows falling particles beingdeposited on the surface giving rise to the noise term, withdown-slope movement of material giving rise to the diffusiveterm. With this model, it is possible to consider if the spec-tral content of the lowlands appears simply as a diffusivelysmoothed version of an initially heavily cratered southernterrain. Regardless of the details, it is expected that any rea-sonable model would predict that short wavelengths evolvefastest, and long wavelengths equilibrate last. For the caseof a diffusive model, simple dimensional analysis with a dif-fusion constant � , the timescale �

�for decay of modes of

wavenumber � will be inversely proportional to ��� . Moreimportantly, in the steady state limit the spectrum shoulddecay as � � [Nattermann and Tang, 1992; Edwards andWilkinson, 1982].

In a frame co-moving with the average height of the sur-face, the continuum limit of the above model is described bythe Edwards-Wilkinson equation [Edwards and Wilkinson,1982], solved in the appendix with the explicit inclusion ofinitial conditions. The evolution in time of an average initial

10 AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY

180˚ 240˚ 300˚ 0˚ 60˚ 120˚ 180˚

-60˚

-30˚

0˚

30˚

60˚

(a)

0 1 2 5 10 20 30 40 50 100 200 500 1000Inter-Quartile Scale (m)

180˚

240˚

300˚

0˚

60˚

120˚180˚180˚

240˚

300˚

0˚

60˚

120˚180˚

80˚N

(b)

180˚

240˚

300˚

0˚

60˚

120˚

180˚180˚

240˚

300˚

0˚

60˚

120˚

180˚

80˚S

(c)

Plate 4. Inter-Quartile Scale (IQS) in 35 km windows. Format is the same as Plate 1. Note the smoothness of AmazonisPlanitia (

� � � N ��� � E).

180˚ 240˚ 300˚ 0˚ 60˚ 120˚ 180˚

-60˚

-30˚

0˚

30˚

60˚

(a)

0 250 500 1000 1500 2000 2500Decorrelation Length l (m)

180˚

240˚

300˚

0˚

60˚

120˚

180˚180˚

240˚

300˚

0˚

60˚

120˚

180˚

80˚N

(b)

180˚

240˚

300˚

0˚

60˚

120˚

180˚180˚

240˚

300˚

0˚

60˚

120˚

180˚

80˚S

(c)

Plate 5. Decorrelation length � in 35 km windows. Format is the same as Plate 1. Olympia Planitia ( � � N � � � � E) dune fieldshave distinctly low decorrelation length. A latitudinal trend shifted slightly northwards is seen, with typical values increasingpolewards by � 50%.

AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY 11

10−2

10−1

100

10−10

10−8

10−6

10−4

10−2

← Region A

β~3.4 →

Region B →

β~1.4 ↑

β~2.0 ↑

↓ β~3.4

Spatial Frequency [cycles/km]

Pow

er [k

m2 ]

110100

Wavelength [km]

Figure 5. Average power spectrum of the topography of anarea in the heavily cratered southern terrain (Region A) andof an area in the northern lowlands (Region B). Note the log-arithmic scale, so that slopes indicate power law exponents.In the short wavelengths the effective exponents for the tworegions are similar, while in the long wavelengths the expo-nent of Region B is lower than that of Region A.

spectrum � ��� � � � , given by Equation A11, is

� ��� ��� � ��� � ��� � ��� � � � � � �/ �

�% � ��� � � ��� � ��

� �� (6)

where the term on the left hand side is the time dependentPSD, � is the diffusion constant,

�is the amplitude of the

noise, � is magnitude of the 2-D wave-vector � , and � is time.Note that as intuition suggested, noisy diffusion affects shortwavelength features (large � in the exponential term) first,and that long wavelength features (small � ) are affected last.Furthermore, in the steady state limit the power law expo-nent predicted in 2-D is ��0 � (where � ��� ��� � ����� ). 1-Dtransects taken from this 2-D field will theoretically have anexponent � � �

[Voss, 1991; Neumann and Forsyth, 1995],but in practice the exponent can be greater ( � 1.2) due to fi-nite domain size effects.

The power-law exponent in the spectral range of wave-lengths 20 to 200 km shown in Figure 5 Region B is con-sistent with evolution of the surface by the above model ofsedimentation. In the decade of wavelengths from 0.7 to 7km the data cannot be explained by noisy diffusion alone,nor in fact by most reasonable models, because they predictfast short-wavelength mode relaxation, and slow relaxationin the long wavelengths. A satisfactory explanation of the

Figure 6. Cartoon showing the growth of a surface under amodel of noisy diffusion.

spectral evolution requires an additional process to steepenthe slope of the short wavelengths. Such a process could, forexample, be roughening by cratering. With an appropriatesize frequency distribution, it can account for the observedpower law exponent � � 3.4 for the high frequency contentof both regions. Since large impactors, and therefore largecratering events are rare, the shallowing of the long wave-length portion of the spectrum from a slope of � 2.1 to � 1.4can be accounted for by a smoothing, diffusion-like processthat acts on a faster timescale than the large craters (thoughstill slower than the short wavelength diffusion).

We therefore conclude that under the assumption of anoisy diffusion model for sedimentation, the lowlands ap-pear as a partially-smoothed version of the south in a limitedwavelength range. The spectral character can be explainedby a diffusive process at long wavelengths, followed by, orconcurrent with, cratering at short wavelengths. The possi-bility that the spectrum of the lowlands was smoothed com-pletely (effectively resetting the surface topography), andthen evolved further to its present shape is not excluded bythis analysis.

Landing Site Selection

Of primary importance for the selection of sites for up-coming Mars landers is the threat posed to the safety of thespacecraft on touchdown, the vehicle’s likely tilt, and possi-bly the maneuverability at the site. Such considerations wereemployed for example in the site selection for Viking Lander2 using radar cross section data. Although the topographicwavelengths most relevant are smaller than the resolution ofMOLA data, the dependence of vertical roughness on scaleis typically monotonic. Therefore MOLA data is a powerfuldiscriminator among surfaces in a comparative sense, whichcan be further calibrated with ground truth.

12 AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY

In Plate 6 we demonstrate the use of MOLA to help char-acterize past landing sites. The top panel (a) shows me-dian slope roughness for the area that was accessible to theMars Polar Lander (MPL) in late August, 1999. The ex-pected current position of the spacecraft is also indicated.At MOLA sampling, smooth plains are prevalent in an areasouth of � S between longitudes

� � � E-� � � � E, but higher

slopes are observed in localized inliers. As shown in Plate6(b) median slopes of the smooth plains in Chryse Planitiahave typical values between

� � � � � � � � (as expected, lowerthan the RMS slope values). The relative smoothness atthis scale is consistent with the surface being a depositionalin origin. Viking Lander 1 and Pathfinder landed in thisarea and provide ground truth for radar [Harmon, 1997] andMOLA-derived roughness at small scales. The Pathfinderlanding site situated at the mouth of Ares Vallis is charac-terized by deposited debris of channelized flows. Golombeket al. [1996] find measured slopes there are consistent withan RMS of � 5 � .

Conclusions

The results discussed here represent a snapshot of an on-going data analysis effort which is progressing both in termsof data volume and coverage, as well as in theory and tech-nique. Thus far, we have demonstrated the advantages ofrobust estimators over traditional ones and have used themto characterize the Martian surface globally. The globalasymmetry in the crust of Mars manifests itself in the bi-modality of the hypsometry, but the elevation distributionfunction becomes unimodal when referenced to the plane-tary COF, similar to Venus but not Earth. Outflow channelsoften follow the direction of long baseline slopes in the to-pography. These tilts are mostly facing north in the northernhemisphere, and in some parts of the south. It was foundthat at MOLA sampling baselines the northern hemisphereis smoother than the south, and that the vast region of Ama-zonis Planitia is unusually smooth. Of the measured ter-restrial planets’ topography, Amazonis is most analogous toheavily-sedimented fluvial basins on Earth such as the oceanfloor. Although this observation represents a piece of cir-cumstantial evidence rather than proof of geologic origin,and despite its very early emergence from MOLA data, ithas potentially tantalizing ramifications and has so far with-stood the tests of repeat measurements and global coverage.

Statistics of topography show that a part of Hellas’s rimhas undergone intensified erosion, consistent with a hypoth-esis of smoothing by an ancient glacier or much larger southpolar cap. We have further shown how the decorrelationlength can be used as a discriminator among geologic sur-faces, and measured a very short correlation length over the

140˚

140˚

160˚

160˚

180˚

180˚

-78˚

-78˚

-76˚

-76˚

-74˚

-74˚

-72˚

-72˚

140˚

140˚

160˚

160˚

180˚

180˚

-78˚

-78˚

-76˚

-76˚

-74˚

-74˚

-72˚

-72˚

MPL

(a)

310˚

310˚

315˚

315˚

320˚

320˚

325˚

325˚

330˚

330˚

20˚ 20˚

25˚ 25˚

Pathfinder

VL1

(b)

0.0 0.1 0.2 0.5 1.0 2.0 5.0 20.0Median Slope (deg)

Plate 6. Median slope roughness maps of (a) area accessi-ble to the Mars Polar Lander and (b) Viking Lander 1 andPathfinder sites in Chryse Planitia.

dunes of Olympia Planitia.

A description of the topography in terms of its powerspectrum demonstrates the difference between the lowlandsand highlands. Modification of the long wavelength por-tions of the north’s spectrum relative to the south’s can beaccounted for by a simple depositional model (i.e. noisydiffusion) such as might be expected at an ocean’s floor.However, the short wavelength spectrum cannot by simulta-neously matched, and its explanation requires an additionalprocess such as cratering.

MOLA slopes and statistical properties of topography arealso being used to characterize potential landing sites for fu-ture Mars landers.

AHARONSON ET AL.: STATISTICS OF MARS’ TOPOGRAPHY 13

Appendix: Power Spectrum Evolution underNoisy Diffusion with Initial Conditions

The effect of many physical processes on the height ofa 2-dimensional interface � ��� ��� � as a function of positionand time can be modeled with the noisy diffusion equation,which can be derived from the continuum limit of a latticemodel of sedimentation [Edwards and Wilkinson, 1982]:�� � ����� ��� � � ��� � ��� � � � �

��� ��� � (A1)

where � is a diffusion constant and��� ��� � is a noisy forcing

term that is uncorrelated in the ensemble average:� ��� � � � � � ��� � � ��� � � ��� ��� � � � � � � � � � (A2)

As have others in the past [e.g. Edwards and Wilkinson,1982; Barabasi and Stanley, 1995; Nattermann and Tang,1992], we seek a spectral domain time-dependent solutionto Equation (A1). Here we explicitly include a prescribedinitial condition so that we may examine the evolution ofone surface from another (B. Newman, pers. comm., 2000).

Rearranging the Edwards-Wilkinson equation (A1), andcorrelating it with itself gives� �� � � � � �� � �� � � � ��� � �� � � ��� � � � � ��� � � � � ���

� � ��� � ��� � � ��� � � ��� � (A3)

Assuming that the field � is stationary in space, the correla-tion function � depends only upon � ��� � , and we define

� ��� ��� � � ����� � � � � � ��� � � � � ��� � � � � ��� � (A4)

Furthermore, � � ��� � � , and we can replace � ��� � with� , then substitute (A2) and (A4) in (A3), to obtain� �� � � ��� �� � �� � � � ��� �� � ��� ����� � � � � � ��� � � � � � � � �(A5)

Multiplication by � � ����� � , and integration over all � , yields� �� � � ��� � � �� � � � � � � � ��� � ��� � � � � � �/ � � � � � � � �

(A6)where � � & � & , and � is defined as

� ��� ������� � � � �� �/ �

�� � ��� ������� � � � � ����� � � � � (A7)

assuming that � decays with & � &���� sufficiently fast thatthe integral converges, i.e. that the Fourier transform ofthe correlation function exists. When � � � � , we identify

� ��� ������� � � � ��� ��� � as the power spectrum, in accordancewith the Wiener-Khinchine theorem [e.g. Gardiner, 1985].

Rearrangement (using an integrating factor) of the left-hand side of equation (A6) yields

� �� �!

� �"$# �� � �

�� � �� � �!

� �"$# � ��� � ����� � � �"� �/ �

�% � � � � � � �(A8)

Multiplying by � � �� �!

� &"'# , integrating from

�to � and

�to� � , and finally taking the particular case � � � � , gives after

some algebra (allowing for both � � � � and � � � � ),� ��� � � � � � � �

� � ��� � ��� � � � � � �

� � ��� � � ��� �

� � � ��� � ��� � � � � � � � �/ �

� ��� � � ��� � ��

� �� � (A9)

By assuming the initial field and the stochastic forcing areuncorrelated, so that for example

� ��� � � � � ��� � � � ��� � �, we

can multiply equation (A1) by � ��� � � � � and repeat the abovesteps to find

� ��� ����� � � � � ��� � � ��� � � � � �� � ��� � � � � � � (A10)

Taking � ��� � � �(� � ��� � � � � � as the spectrum of the initialfield, and substituting (A10) in (A9), the time-dependentpower spectrum is:

� ��� ��� � ��� � ��� � ��� � � � � � �/ �

� ��� � � ��� � ��

� � �(A11)

Acknowledgments. We thank John Goff and Bill Newmanfor careful reviews, and acknowledge helpful discussions with GregNeumann and Jim Head. It was our pleasure to incorporate com-ments by Bill Newman on an early draft of this paper, which helpedformulate the appendix in a more precise manner. This study wassupported by the generous Kerr Fellowship, the Mars Global Sur-veyor Project, and by DOE Grant DE FG02-99ER 15004.

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O. Aharonson, M. T. Zuber, and D. H. Rothman, Depart-ment of Earth, Atmospheric and Planetary Sciences, Mas-sachusetts Institute of Technology, Cambridge, MA 02139(e-mail: oded@mit.edu)Received September 27, 2000; accepted January 11, 2001.